Bones getting left behind in mesh transform
So what you could do is:


First, to clean a bit, remove the current armature and the object Armature modifier.





Also, delete all the existing vertex groups.





Prepare the right side of the armature, each bone must be the child of another one, the main one is the root bone. Put their head on the paper fold so that it will fold correctly. Give them some logical names fold Wing1_R, etc... Once you're good, select them all and in the header menu select Armature > Symmetrize, it will mirror your bones (as long as they have the R or L suffix).





Parent the paper to the armature With Empty Groups, which means some vertex groups will be created within the object, they will have the name of the bones, but they won't be affected to any bone yet. It's the best way to parent as your object is not organic.





Now select some vertices and in the Vertex Groups list, assign them to the group to which there are supposed to be part of, meaning which have the name of the bone that is supposed to influence them.




Blend file here: